Talk:Diophantine equation

[Untitled]
tell me in the way to figure out the problem on his tomb theat tells how old he was when he died... please tell me how to do that problem

See here: Diophantus Tombstone Problem

Fibonacci Sequence and its Second Degree Diophantine Equation
Forty years ago I discovered that the Fibonacci Sequence (1, 1, 2, 3, 5, 8, etc) can be generated from the second degree Diophantine equation

5k^2 -/+ 4 = m^2   where the -,+ is taken alternately.

The equation is a variation on Pell's, in that

x^2 - ny^2 = +/- 4   instead of 1.

Walter Abetz 13:32, 19 November 2006 (UTC)

Arithmetic Geometry
I'm personally not clear on the distinction between Arithmetic Geometry and Algebraic Geometry, except some vague awareness that the former usually deals with rational points on varieties and the latter with more general algebraic structure. So I can kind of see why 'Arithmetic geometry' redirects to this page, since finding rational points on varieties is roughly speaking the same problem as solving Diophantine equations. But it'd be nice if someone who knows something about the subject could insert a paragraph or two on this page about it, if not create an actual page for it. Chenxlee (talk) 17:55, 11 February 2008 (UTC)

Markov Diophantine Equations
Does someone more qualified want to discuss these? I'm only a high school student.

Inconsistency in the article
In the beginning of the article a Diophantine equation is defined to be a _polynomial_ equation (with certain properties and conditions), but one of the given examples, namely 4/n = 1/x + 1/y + 1/z, is in fact not a polynomial equation by the definition given in the article about "polynomial" because it contains division. Please someone fix that. Either change the definiton of an Diophantine equation or remove the offending example. (I dont know which is the correct way) —Preceding unsigned comment added by 213.240.234.31 (talk) 19:58, 11 November 2008 (UTC)


 * The example is trivially modifiable to be a polynomial: 4xyz = yzn + xzn + xyn. So it's not really a violation of the definition, but maybe some explanation of why it isn't might be useful. —David Eppstein (talk) 20:03, 11 November 2008 (UTC)


 * Thank you. (Now i'm happy:)) —Preceding unsigned comment added by 213.240.234.31 (talk) 20:14, 11 November 2008 (UTC)

Definition reads oddly
"...[a polynomial equation] that allows two or more variables to take integer values only."

Isn't this saying the same thing as "...that restricts two or more variable to integer values only", and if so, isn't the second wording clearer? 74.192.47.171 (talk) 05:21, 25 December 2013 (UTC)jt
 * — D.Lazard (talk) 12:02, 25 December 2013 (UTC)

To D.Larazd
If you actually look at the submission date of the "1980" paper it was submitted on May 16, 1978. The 1979 Fibonacci Quarterly paper is also cited in Schrijver's book: The former produces smaller solutions. The "1980" (actually 1978) paper says:

"To illustrate the discussion, consider a small example with n=3 and a=(8913, 5677, 4378). The algorithm with Rule A produces generating vectors (0, 4378, -5677) and (1, 1, 12736677, -16515789), whereas the procedure with Rule B gives (8189939, -227499, -16378578) and (989097, -27475, -1978037). In general, the application of Rule B does not reduce the size of the elements in the generating vectors."

The 1979 paper says:

"For illustrative purposes, we will continuously use the following example with n = 3: (al, a2, a3) = (8913, 5677, 4378). Or, we are interested in the generator of: 8913x1 + 5677x2 + 4378x3 = 0. It turns out that the Bond Algorithm [3] produces the two generating vectors (5677, -8913, 0) and (2219646, 3484888, -1), whereas the procedure we propose gives (cf. Section 3) (-57, 17, 94) and (61, -95, -1)"

The fact that neither paper cites the other is a bit amusing, but which is the notable one is clear from both their text. 86.127.138.234 (talk) 16:36, 13 February 2015 (UTC)

Fell free to add the other (1980) paper if you think people should read it :-) The only more recent paper on the topic cited in Schrijver's book is a 1981 paper by Kerztner in AMM. However, I would not suggest adding Kerztner's paper because it turned out unoriginal, i.e. the same as Blankinship's algorithm; a note to that effect was published in AMM in 1983 . 86.127.138.234 (talk) 17:15, 13 February 2015 (UTC)
 * I have used the non-notability for removing your edit because it was the simplest argument. In fact the reason of my revert is more complex. Firstly, I agree that the section "One equation" should be renamed "one equation in two unknowns", and that a section "One equation in more unknowns" could be added. However, Wikipedia is an encyclopedia, and if a subject is treated, its coverage must not be reduced to a single 35 years old paper, ignoring the more recent results on the subject. Therefore including this article in "See also" section gives a undue weight to this particular paper. Since 1980, the main progresses that I know of are
 * The results on complexity and practical efficiency for the computation of the Smith normal form, which contain bounds on the size of the output and of the intermediate integers. They give thus a quantitative version of the main objective of the paper you have cited, which makes it out of date, I believe.
 * The LLL algorithm, witch is very fast for finding a solution with small coefficients from any general solutions.
 * In my opinion, this article deserve a section "One equation in more unknowns", but it must not be based on a single primary source. D.Lazard (talk) 18:58, 13 February 2015 (UTC)
 * Get to work then and add these algorithms (you are so sure exist) to the article on Smith normal form; all I see there is a paper from 1861 (I didn't get the century wrong.) 86.127.138.234 (talk) 14:00, 14 February 2015 (UTC)

One linear equation with more than two variables
I think the article should mention how to expand the solution from two variables to more. Here is a helpful link: https://math.stackexchange.com/questions/20906/how-to-find-an-integer-solution-for-general-diophantine-equation-ax-by-cz — Preceding unsigned comment added by W3ricardo (talk • contribs) 04:29, 15 April 2018 (UTC)
 * The general case of linear Diophantine equations (any number of unknowns, any number of equations) is treated two subsections later. D.Lazard (talk) 08:52, 15 April 2018 (UTC)

infinite equation
even if the variables were all simply one, wouldn't this be infinite? unless zeros are allowed. --142.163.195.48 (talk) 20:46, 26 January 2021 (UTC)

"Sums of three cubes"
Should Sums of three cubes be included in the 'see also' section? somewhat popular among the numberphile crowd. --142.163.195.48 (talk) 20:58, 26 January 2021 (UTC)

Simon's Favorite Factoring Trick
Maybe we should have $$axy + bx + cy = d$$ in the examples of diophantine equations. I believe it's is a good example because it is both common and easy to solve—using SFFT ("Simon's Favorite Factoring Trick"): $$ \begin{align} cxy + ax + by & = d \\ \Longleftrightarrow (cx + b)(cy + a) & = ab + cd \end{align} $$ Now you can consider the factors of $$ab + cd$$ to solve for $$(x, y)$$. Revol Ufiaw (talk) 09:55, 29 May 2024 (UTC)